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# Functional Equations

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Let $$S$$ be the set of nonzero real numbers. Let $$f : S \to \mathbb{R}$$ be a function such that

(i) $$f(1) = 1$$

(ii) $$f \left( \frac{1}{x + y} \right) = f \left( \frac{1}{x} \right) + f \left( \frac{1}{y} \right)$$ for all $$x, y \in S$$  such that $$x + y \in S$$ and

(iii)$$(x + y) f(x + y) = xyf(x)f(y)$$for all $$x, y \in S$$ such that $$x + y \in S$$

Find the number of possible functions $$f(x)$$.

From experimenting and algebra, $$f(x)=\frac{1}{x}$$ is a valid solution.  I just want to know if there are any more.  If I get the question right, the website will show me an explanation of the solution, so feel free to leave out complex LaTeX formatting if you want.

Jun 20, 2021